"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

Posts Tagged ‘cohomology’

Yesterday we gave a brief and abstract description of Galois descent, the punchline of which was that Galois descent could abstractly be described as a natural equivalence

where is a Galois extension, is the Galois group of (thinking of as an object of the category of field extensions of at all times), is a category of “objects over ,” and is a category of “objects over .”

In fact this description is probably only correct if is a finite Galois extension; if is infinite it should probably be modified by requiring that every function of that occurs (e.g. in the definition of homotopy fixed points) is continuous with respect to the natural profinite topology on . To avoid this difficulty we’ll stick to the case that is a finite extension.

Today we’ll recover from this abstract description the somewhat more concrete punchline that -forms of an object can be classified by Galois cohomology , and we’ll give some examples.

Yesterday we described how a (finite-dimensional) projective representation of a group functorially gives rise to a -linear action of on such that the Schur class classifies this action.

Today we’ll go in the other direction. Given an action of on explicitly described by a 2-cocycle , we’ll recover the category of -projective representations, or equivalently the category of modules over the twisted group algebra, by taking the homotopy fixed points of this action. We’ll end with another puzzle.

Today we’ll resolve half the puzzle of why the cohomology group appears both when classifying projective representations of a group over a field and when classifying -linear actions of on the category of -vector spaces by describing a functor from the former to the latter.

Three days ago we stated the following puzzle: we can compute that isomorphism classes of -linear actions of a group on the category of vector spaces over a field correspond to elements of the cohomology group

.

This is the same group that appears in the classification of projective representations of over , and we asked whether this was a coincidence.

Before answering the puzzle, in this post we’ll provide some relevant background information on projective representations.

Previously we described what it means for a group to act on a category (although we needed to slightly correct our initial definition). Today, as the next step in our attempt to understand Galois descent, we’ll describe what the fixed points of such a group action are.

John Baez likes to describe (vertical) categorification as replacing equalities with isomorphisms, which we saw on full display in the previous post: we replaced the equality with isomorphisms , and as a result we found 2-cocycles lurking in this story.

I prefer to describe categorification as replacing properties with structures, in the nLab sense. That is, the real import of what we just did is to replace the property (of a function between groups, say) that with the structure of a family of isomorphisms between and . The use of the term “structure” emphasizes, as we also saw in the previous post, that unlike properties, structures need not be unique.

Accordingly, it’s not surprising that being a fixed point of a group action on a category is also a structure and not a property. Suppose is a group action as in the previous post, and is an object. The structure of a fixed point, or more precisely a homotopy fixed point, is the data of a family of isomorphisms

which satisfy the compatibility condition that the two composites

and

are equal, as well as the unit condition that

where is the unit isomorphism . This is, in a sense we’ll make precise below, a 1-cocycle condition, but this time with nontrivial (local) coefficients.

Curiously, when the action is trivial (meaning both that and that ), this reduces to the definition of a group action of on in the usual sense. In general, we can think of homotopy fixed point structure as a “twisted” version of a group action on where the twist is provided by the group action on .

Yesterday we decided that it might be interesting to describe various categories as “fixed points” of Galois actions on various other categories, whatever that means: for example, perhaps real Lie algebras are the “fixed points” of a Galois action on complex Lie algebras. To formalize this we need a notion of group actions on categories and fixed points of such group actions.

So let be a group and be a category. For starters, we should probably ask for a functor for each . Next, we might naively ask for an equality of functors

but this is too strict: functors themselves live in a category (of functors and natural transformations), and so we should instead ask for natural isomorphisms

.

These natural isomorphisms should further satisfy the following compatibility condition: there are two ways to use them to write down an isomorphism , and these should agree. More explicitly, the composite

should be equal to the composite

.

(There’s also some stuff going on with units which I believe we can ignore here. I think we can just require that on the nose and nothing will go too horribly wrong.)

These natural isomorphisms can be regarded as a natural generalization of 2-cocycles, and the condition above as a natural generalization of a cocycle condition. Below the fold we’ll describe this and other aspects of this definition in more detail, and we’ll end with two puzzles about the relationship between this story and group cohomology.

The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let be a commutative ring, which in the most classical case might be a number field like or the ring of integers in a number field like . Suppose we want to find solutions, over , of a system of polynomial equations

.

Then it’s not hard to see that this problem is equivalent to the problem of finding -algebra homomorphisms from to . This is equivalent to the problem of finding left inverses to the morphism

of commutative rings making an -algebra, or more geometrically equivalent to the problem of finding right inverses, or sections, of the corresponding map

of affine schemes. Allowing to be a more general scheme over can also capture more general Diophantine problems.

The problem of finding sections of a morphism – call it the section problem – is a problem that can be stated in any category, and the goal of this post is to say some things about the corresponding problem for spaces. That is, rather than try to find sections of a map between affine schemes, we’ll try to find sections of a map between spaces; this amounts, very roughly speaking, to solving a “topological Diophantine equation.” The notation here is meant to evoke a particularly interesting special case, namely that of fiber bundles.

We’ll try to justify the section problem for spaces both as an interesting problem in and of itself, capable of encoding many other nontrivial problems in topology, and as a possible source of intuition about Diophantine equations. In particular we’ll discuss what might qualify as topological analogues of the Hasse principle and the Brauer-Manin obstruction.